Two hundred and 5, point zero nine or 0 9
Answer: 1. y = 2(x + 4)² - 3
![\bold{2.\quad y=-\dfrac{1}{3}(x+8)^2-7}](https://tex.z-dn.net/?f=%5Cbold%7B2.%5Cquad%20y%3D-%5Cdfrac%7B1%7D%7B3%7D%28x%2B8%29%5E2-7%7D)
![\bold{3.\quad y=-\dfrac{1}{2}(x-7)^2+1}](https://tex.z-dn.net/?f=%5Cbold%7B3.%5Cquad%20y%3D-%5Cdfrac%7B1%7D%7B2%7D%28x-7%29%5E2%2B1%7D)
<u>Step-by-step explanation:</u>
Notes: The vertex form of a parabola is y = a(x - h)² + k
- (h, k) is the vertex
- p is the distance from the vertex to the focus
![\bullet\quad a=\dfrac{1}{4p}](https://tex.z-dn.net/?f=%5Cbullet%5Cquad%20a%3D%5Cdfrac%7B1%7D%7B4p%7D)
1)
![\text{Vertex}=(-4,-3)\qquad \text{Directrix}:y=-\dfrac{25}{8}\\\\\text{Given}:(h, k)=(-4, 3)\\\\p=\dfrac{-24}{8}-\dfrac{-25}{8}=\dfrac{1}{8}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{1}{8})}=\dfrac{1}{\frac{1}{2}}=2](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%28-4%2C-3%29%5Cqquad%20%5Ctext%7BDirectrix%7D%3Ay%3D-%5Cdfrac%7B25%7D%7B8%7D%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%28-4%2C%203%29%5C%5C%5C%5Cp%3D%5Cdfrac%7B-24%7D%7B8%7D-%5Cdfrac%7B-25%7D%7B8%7D%3D%5Cdfrac%7B1%7D%7B8%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B1%7D%7B8%7D%29%7D%3D%5Cdfrac%7B1%7D%7B%5Cfrac%7B1%7D%7B2%7D%7D%3D2)
Now input a = 2 and (h, k) = (-4, -3) into the equation y = a(x - h)² + k
y = 2(x + 4)² - 3
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2)
![\text{Vertex}=(-8,-7)\qquad \text{Directrix}:y=-\dfrac{-25}{4}\\\\\text{Given}:(h, k)=(-8, -7)\\\\p=\dfrac{-28}{4}-\dfrac{-25}{4}=\dfrac{-3}{4}}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-3}{4})}=\dfrac{1}{-3}=-\dfrac{1}{3}](https://tex.z-dn.net/?f=%5Ctext%7BVertex%7D%3D%28-8%2C-7%29%5Cqquad%20%5Ctext%7BDirectrix%7D%3Ay%3D-%5Cdfrac%7B-25%7D%7B4%7D%5C%5C%5C%5C%5Ctext%7BGiven%7D%3A%28h%2C%20k%29%3D%28-8%2C%20-7%29%5C%5C%5C%5Cp%3D%5Cdfrac%7B-28%7D%7B4%7D-%5Cdfrac%7B-25%7D%7B4%7D%3D%5Cdfrac%7B-3%7D%7B4%7D%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B-3%7D%7B4%7D%29%7D%3D%5Cdfrac%7B1%7D%7B-3%7D%3D-%5Cdfrac%7B1%7D%7B3%7D)
Now input a = -1/3 and (h, k) = (-8, -7) into the equation y = a(x - h)² + k
![\bold{y=-\dfrac{1}{3}(x+8)^2-7}](https://tex.z-dn.net/?f=%5Cbold%7By%3D-%5Cdfrac%7B1%7D%7B3%7D%28x%2B8%29%5E2-7%7D)
******************************************************************************************
3)
![\text{Focus}=\bigg(7,\dfrac{1}{2}\bigg)\qquad \text{Directrix}:y=\dfrac{3}{2}](https://tex.z-dn.net/?f=%5Ctext%7BFocus%7D%3D%5Cbigg%287%2C%5Cdfrac%7B1%7D%7B2%7D%5Cbigg%29%5Cqquad%20%5Ctext%7BDirectrix%7D%3Ay%3D%5Cdfrac%7B3%7D%7B2%7D)
The midpoint of the focus and directrix is the y-coordinate of the vertex:
![\dfrac{focus+directrix}{2}=\dfrac{\frac{1}{2}+\frac{3}{2}}{2}=\dfrac{\frac{4}{2}}{2}=\dfrac{2}{2}=1](https://tex.z-dn.net/?f=%5Cdfrac%7Bfocus%2Bdirectrix%7D%7B2%7D%3D%5Cdfrac%7B%5Cfrac%7B1%7D%7B2%7D%2B%5Cfrac%7B3%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B%5Cfrac%7B4%7D%7B2%7D%7D%7B2%7D%3D%5Cdfrac%7B2%7D%7B2%7D%3D1)
The x-coordinate of the vertex is given in the focus as 7
(h, k) = (7, 1)
Now let's find the a-value:
![p=\dfrac{2}{2}-\dfrac{3}{2}=\dfrac{-1}{2}}\\\\\\a=\dfrac{1}{4p}=\dfrac{1}{4(\frac{-1}{2})}=\dfrac{1}{-2}=-\dfrac{1}{2}](https://tex.z-dn.net/?f=p%3D%5Cdfrac%7B2%7D%7B2%7D-%5Cdfrac%7B3%7D%7B2%7D%3D%5Cdfrac%7B-1%7D%7B2%7D%7D%5C%5C%5C%5C%5C%5Ca%3D%5Cdfrac%7B1%7D%7B4p%7D%3D%5Cdfrac%7B1%7D%7B4%28%5Cfrac%7B-1%7D%7B2%7D%29%7D%3D%5Cdfrac%7B1%7D%7B-2%7D%3D-%5Cdfrac%7B1%7D%7B2%7D)
Now input a = -1/2 and (h, k) = (7, 1) into the equation y = a(x - h)² + k
![\bold{y=-\dfrac{1}{2}(x-7)^2+1}](https://tex.z-dn.net/?f=%5Cbold%7By%3D-%5Cdfrac%7B1%7D%7B2%7D%28x-7%29%5E2%2B1%7D)
Step-by-step explanation:
The Circumference is:
![c = 2\pi \: r](https://tex.z-dn.net/?f=c%20%3D%202%5Cpi%20%5C%3A%20r)
Radius r is:
![r = \frac{d}{2}](https://tex.z-dn.net/?f=r%20%3D%20%20%5Cfrac%7Bd%7D%7B2%7D%20)
So,
![c = \pi \: d \\ d = \frac{c}{\pi} \\ d = \frac{604}{\pi} \: cm](https://tex.z-dn.net/?f=c%20%3D%20%5Cpi%20%5C%3A%20d%20%5C%5C%20d%20%3D%20%20%5Cfrac%7Bc%7D%7B%5Cpi%7D%20%20%5C%5C%20d%20%3D%20%20%5Cfrac%7B604%7D%7B%5Cpi%7D%20%5C%3A%20%20cm)
Answer:
$45???
Step-by-step explanation:
Answer:
20 + 2 =22 or in words it would be twenty two
Step-by-step explanation: